Skewed distributions

What Are Skewed Distributions?

Skewed distributions refer to probability distribution patterns that are not symmetrical around their central tendency. In statistics and financial analysis, a distribution is considered skewed if its values are concentrated on one side, with a longer "tail" extending to the other. This asymmetry indicates that data points are not evenly distributed around the mean. Unlike a normal distribution, which is perfectly symmetrical and often represented by a bell curve, skewed distributions suggest the presence of extreme values or outliers pulling the distribution in one direction.

History and Origin

The mathematical understanding and quantification of statistical distributions, including their asymmetry, evolved significantly in the late 19th and early 20th centuries. A pivotal figure in this development was English statistician Karl Pearson. Pearson, often credited with establishing the discipline of mathematical statistics, introduced foundational concepts such as moments, which are used to describe the shape of a data set. He devised a system of continuous probability distributions, now known as Pearson distributions, specifically to model asymmetrical (skewed) patterns observed in various forms of data.⁸ His work provided a robust framework for quantifying the degree and direction of skewness.⁶, ⁷

Key Takeaways

Skewed distributions are asymmetrical, meaning data points are not evenly distributed around the mean.
Positive skew (right-skewed) indicates a long tail on the right, with the mean typically greater than the median.
Negative skew (left-skewed) indicates a long tail on the left, with the mean typically less than the median.
Understanding skewness is crucial for proper risk assessment and financial modeling, as it highlights the presence of extreme values.
It influences the choice of appropriate statistical methods for data analysis.

Formula and Calculation

The most common method for calculating skewness is Pearson's moment coefficient of skewness, also known as the third standardized moment. For a sample, the formula is:

\text{Skewness} = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s} \right)^3

Where:

(n) = the number of data points in the sample
(x_i) = the (i)-th data point
(\bar{x}) = the mean of the sample
(s) = the standard deviation of the sample

Another common measure is Pearson's first coefficient of skewness, which relates the mean, mode, and standard deviation:

\text{Skewness}_1 = \frac{\bar{x} - \text{Mode}}{s}

And Pearson's second coefficient of skewness, often used when the mode is not well-defined, uses the median:

\text{Skewness}_2 = \frac{3(\bar{x} - \text{Median})}{s}

Interpreting Skewed Distributions

The numerical value of skewness indicates both the direction and magnitude of the asymmetry:

Positive Skew (Right-Skewed): A positive skewness value means the tail of the distribution extends towards the right, with a higher concentration of data on the left side. In such cases, the mean is typically greater than the median, which is often greater than the mode ((\text{Mean} > \text{Median} > \text{Mode})). This pattern suggests a few large positive values are pulling the mean higher.
Negative Skew (Left-Skewed): A negative skewness value means the tail extends towards the left, with most data concentrated on the right side. Here, the mean is typically less than the median, which is often less than the mode ((\text{Mean} < \text{Median} < \text{Mode})). This indicates the presence of a few significantly small negative values dragging the mean lower.
Zero Skew: A skewness of zero indicates a perfectly symmetrical distribution, where the mean, median, and mode are all equal.

Understanding the skewness of a data set provides valuable insights beyond simply knowing the average or spread. It helps in recognizing the characteristic shape of the data and identifying the presence of extreme observations.

Hypothetical Example

Consider a hypothetical distribution of annual asset returns for two different investments, A and B, over several years.

Investment A (Positive Skew):
Returns: -5%, 2%, 3%, 4%, 5%, 6%, 7%, 8%, 20%

Mean: Approximately 5.56%
Median: 5%
Mode: No distinct mode
The large return of 20% pulls the mean higher than the median, creating a positive skew. An investor might see this as appealing due to the potential for large positive surprises, though such events are rare.

Investment B (Negative Skew):
Returns: -20%, -5%, 2%, 3%, 4%, 5%, 6%, 7%, 8%

Mean: Approximately 1.11%
Median: 4%
Mode: No distinct mode
The significant negative return of -20% pulls the mean lower than the median, creating a negative skew. This indicates that while average returns might seem acceptable, there's a risk of substantial downside events. For portfolio performance evaluation, this negative skew is a critical factor to consider, highlighting potential tail risks.

Practical Applications

Skewed distributions appear frequently in market data and financial analysis. Recognizing and accounting for skewness is essential in various financial contexts:

Asset Returns: Distributions of asset returns often exhibit skewness. For instance, equity returns are frequently observed to be negatively skewed, meaning there's a higher probability of small gains and a smaller, but more impactful, probability of large losses. This contrasts with a normal distribution assumption, where gains and losses are symmetrically probable. Researchers have explored whether skewness persists in common stock returns over time.⁵
Risk Management: Understanding the skewness of potential outcomes is vital for risk assessment. A negatively skewed distribution of portfolio returns, for example, indicates a greater chance of significant downside events than a positively skewed one. Financial professionals use this information to calculate measures like Value-at-Risk (VaR) more accurately, especially when dealing with non-normal distributions.
Option Pricing: Models for pricing options often incorporate skewness and kurtosis to better reflect actual market behavior. The "implied volatility skew" observed in options markets directly reflects the market's perception of the likelihood of extreme price movements.
Economic Indicators: Macroeconomic data, such as income distribution or certain labor market statistics, can also exhibit skewness. For example, some economic metrics observed by central banks, like GDP growth, can be skewed by unusual events such as large tariff-driven import shifts.⁴ The Federal Reserve also monitors "policy rate skew" to gauge market perceptions of future interest rate risks.³
Quantitative Analysis: In quantitative analysis, skewness helps analysts choose appropriate statistical tests and models. Many classical statistical tests assume normally distributed data; if data are significantly skewed, alternative non-parametric tests or data transformations may be necessary to ensure valid conclusions.

Limitations and Criticisms

While a valuable descriptive statistic, skewness has its limitations:

Sensitivity to Outliers: Skewness is highly sensitive to extreme outliers. A single, unusually large or small value can significantly distort the skewness measure, making it less representative of the overall distribution's shape, especially in smaller data sets.²
Sample Size Dependency: For small sample sizes, the calculated skewness may not accurately reflect the true skewness of the underlying population distribution. As sample sizes increase, the sample skewness tends to converge to the population skewness.
Bias in Estimators: Conventional estimators of skewness, particularly those based on the third central moment, can yield biased values in non-normal distributions. This means the calculated skewness might systematically deviate from the true underlying skewness of the distribution.¹
Incomplete Picture: Skewness describes only one aspect of a distribution's shape—its asymmetry. It does not provide information about the "tailedness" or peakedness of the distribution, which is described by kurtosis. A complete understanding of a distribution requires evaluating both measures.
Misinterpretation: A common pitfall is to interpret any non-zero skewness as a definitive sign of a non-normal distribution. While true for perfect symmetry, some distributions may have slight skewness but still be considered approximately normal for practical purposes. Context and other statistical tests are crucial.

Skewed Distributions vs. Normal Distribution

The primary distinction between skewed distributions and a normal distribution lies in their symmetry.

Feature	Skewed Distributions	Normal Distribution
Symmetry	Asymmetrical; data concentrated on one side.	Perfectly symmetrical around the mean.
Tail Behavior	One tail is longer and heavier than the other.	Tails are equal in length and weight.
Mean, Median, Mode	Typically, Mean ≠ Median ≠ Mode (or some inequality).	Mean = Median = Mode.
Shape	Can be positive (right-skewed) or negative (left-skewed).	Bell-shaped (Gaussian curve).
Extreme Values	Indicates presence of more extreme values on one side.	Extreme values are equally likely on both sides.

While a normal distribution is a theoretical ideal, many real-world financial and economic data sets exhibit some degree of skewness. Recognizing this difference is crucial for accurately assessing risk, making forecasts, and applying appropriate analytical models.

FAQs

What does positive skewness mean in finance?

In finance, positive skewness means that a distribution of asset returns has a longer tail on the right side. This indicates a higher frequency of small losses or gains, but a rare chance of very large positive gains. Investors sometimes prefer positively skewed returns, as it suggests the potential for "home run" investments, even if average returns are modest.

What does negative skewness mean in finance?

Negative skewness in finance signifies that the distribution of returns has a longer tail on the left side, indicating a higher frequency of small gains and a rare chance of very large negative losses. This is often observed in stock markets, where sharp, sudden drops (crashes) occur less frequently but are more severe than typical daily fluctuations. Understanding this helps in risk assessment, as it highlights potential downside risk.

How does skewness affect investment decisions?

Skewness significantly impacts investment decisions by revealing the nature of potential extreme outcomes. A fund with negatively skewed returns might appear attractive based on average returns and volatility, but its negative skew warns of greater downside risk. Conversely, an investment with positive skew might be appealing for its upside potential. Investors seeking to avoid large losses might prioritize investments with less negative skew, even if it means sacrificing some average return. This goes beyond simple mean-variance analysis to consider the entire shape of the return distribution.